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3
votes
2answers
359 views

volume of complex hyperbolic manifolds

I would like to know if there are in the literature explicit computations of the volume of complex hyperbolic manifolds. More precisely, let $\mathcal O$ be an imaginary quadratic number field, and ...
2
votes
1answer
167 views

Example of hyperbolic 3-fold with no embedded incompressible subsurfaces

Kahn-Markovic show that every hyperbolic 3-fold contains an immersed $\pi_1$ injective surface. Are there any known examples of hyperbolic 3-folds that do not contain a embedded $\pi_1$ injective ...
2
votes
2answers
306 views

Hurwitz's automorphisms theorem for infinite genus Riemann surfaces

Hurwitz's automorphisms theorem states that for a compact Riemann surface $X$ the cardinality of $Aut(X)$, the group of holomorphic automorphisms, is bounded above by $84(g(X)-1)$ and is therefore ...
9
votes
3answers
626 views

Primitive elements in a free group of rank three

It is well-known that the fundamental group of a twice-punctured torus is a free group of rank three. I see that there is no one-to-one correspondence between the homotopy classes of essential ...
3
votes
3answers
906 views

Poincare Metric on Hyperbolic Plane

as is well known, we can put a metric on the upper half plane $\mathbb{R}^+ \times \mathbb{R}$ by setting $$ d\left((x,t);(x',t')\right):=\log\left(\frac{1 + \delta}{1 - \delta}\right)^{1/2}, $$ ...
12
votes
2answers
640 views

How to see isometries of figure 8 knot complement

The figure 8 knot complement $M$ is the orientable double cover of the Gieseking manifold, which implies that $M$ has a fixed-point free involution. If we think of $M$ with its hyperbolic metric, this ...
2
votes
1answer
753 views

How to rigorously prove that simple closed curves on a surface are primitive closed curves ?

Let me first state the definitions : A not-nullhomotopic closed curve / loop $c$ on an orientable surface $X,c:[0,1]\to X$ is called simple closed curve is $c|[0,1)$ is injective and [ $c(0)=c(1) ] ...
0
votes
1answer
495 views

Is there a similar formula in spherical and hyperbolic geometry as Euclidean Geometry? [closed]

In an Euclidean plane, we know that the area of a triangle is determined by the length of base and the height, i.e., $$ S_{\Delta}=\frac{1}{2}a.h, $$ where $a$ is the length of base and the $h$ is ...
4
votes
1answer
298 views

Arithmetic Fuchsian group

Dear all, I have the following questions: Are all Fuchsian groups of signature $(0;2,2,2,\infty)$ arithmetic? What is known about the trace fields of these groups? Best, K.
3
votes
2answers
245 views

In hyperbolic 3-orbifold with totally geodesic boundary case, is it true: rank(the fundamental group of boundary M)< or equal 2 rank(fundmental group of M)?

For a orientable three manifold M with totally geodesic boundary, this inequality is true. Because the rank of (fundemantal group of boundary M)=rank (homology group of boundary M ) then we use the ...
1
vote
1answer
103 views

Will the rank of fundemantal 3 manifold be decreased is I module the n(n>1) times of a element?

I am doing some rank control about the fundamental group of a 3 dim hyperbolic orbifold. After cuting out the regular neighborhood of all the singularity, I get a manifold with imcompressible ...
4
votes
1answer
421 views

The geometrical meaning of the common value in the law of sines in hyperbolic geometry

What is the geometrical meaning of the common value in the law of sines, $\frac{\sin A}{\sinh a} = \frac{\sin B}{\sinh b} = \frac{\sin C}{\sinh c}$ in hyperbolic geometry? I know the meaning of this ...
8
votes
2answers
831 views

Questions on Thurston's earthquake flow

Here are some questions about the earthquake deformation of hyperbolic surface that I can't answer or find references. I briefly recall the settings. Let's fix a closed surface $S$ with genus $g\geq ...
6
votes
0answers
201 views

Does every hyperbolic 3 manifold with totally geodesic boundary has some finite covering space with more than one boundary component?

I am thinking about the question that: if we double a hyperbolic 3 manifold along its boundary, will the rank of fundemental group of the resulting closed manifold be strictly larger than before?\ The ...
6
votes
1answer
261 views

Can most 3 dimensional hyperbolic orbifolds with finite volume be covered by a hyperbolic manifold?

If G is a discrete cofinite volume subgroup of PSL(2,C),then G acts on H3, H3/G is a 3-dim hyperbolic orbifold N with finite volume, my question is : Is it right in most situations that we can find a ...
5
votes
2answers
282 views

Optimal pants decompositions of a hyperbolic surface

Let $S$ be a hyperbolic surface, which is not the punctured torus or $4$-holed sphere. I am interested in finding a ``geometrically optimal'' pants decomposition on $S$. Here is a candidate ...
2
votes
1answer
413 views

The smallest positive eigenvalue and the length of the shortest geodesic

I'm confused about some things concerning lengths of geodesics on Riemann surfaces and positive eigenvalues of the Laplacian. Moreover, I'm also interested in the relation between these two. Let $X$ ...
0
votes
1answer
205 views

Homotopy Groups of de Sitter and Anti-de Sitter?

Given $n$-dimensional de Sitter or Anti-de Sitter space, $dS_n$ or $AdS_n$, what are the homotopy groups $\pi_m(dS_n)$ and $\pi_m(AdS_n)$ and how does one calculate such things?
3
votes
1answer
436 views

Connection 1-forms of a Riemannian metric and the norm of the Hessian and ( seemingly ) two different definitions of Hessian and its norm

In the paper "On Quasiconformal Harmonic Maps " (link here) by L. F. Tam and T.Y.H. Wan, Pacific Journal of Mathematics, vol 182, no 2, 1998, in section 1, they define the Hessian of a function $f ...
1
vote
0answers
216 views

A regularity question on the Beltrami equation $ f_\bar{z} =\mu . f_z$ on $D$

Hello, This question is related to Chapter V, lemma 3 on page 54 of Lars Ahlfors' 'Lectures on Quasiconformal mappings' which states : If $\mu:\mathbb{C}\to \mathbb{D} \in W^{1,p}(\mathbb{C}), p ...
7
votes
3answers
957 views

Hyperbolicity on Riemann Surfaces

For Riemann surfaces there are at least to possible notions of hyperbolicity. The classical one given by the Uniformization Theorem, or equivalently the type problem, which essentially says that a ...
2
votes
1answer
632 views

A question on part of “An introduction to teichmuller spaces” by Imayoshi-Taniguchi

I was reading the part on Fenchel-Nielsen coordinates, the proof on page 64, I don't understand when they say: Since $\theta_j(t_1)=\theta_j(t_2)$, $j=1,\ldots,3g-3$, all $g_k$ can be glued ...
6
votes
1answer
260 views

What is the complex structure on the boundary torus of a hyperbolic knot complement?

Let $K$ be a hyperbolic knot in $\mathbb S^3$. Restrict the corresponding representation $\pi_1(\mathbb S^3\setminus K)\to\operatorname{PSL}(2,\mathbb C)$ to the fundamental group of the boundary ...
8
votes
0answers
349 views

Is there a general dilogarithm formula for the Cheeger--Chern--Simons class?

I'm looking for a generalization of the calculation of the hyperbolic volume and Chern--Simons invariant for $\operatorname{SL}(2,\mathbb C)$ representations in terms of the Rogers dilogarithm. ...
17
votes
3answers
740 views

Hyperbolic Coxeter polytopes and Del-Pezzo surfaces

Added. In the following link there is a proof of the observation made in this question: http://dl.dropbox.com/u/5546138/DelpezzoCoxeter.pdf I would like to find a reference for a beautiful ...
7
votes
1answer
441 views

Laminations as a limit of ideal triangulations

I am wondering about the following: Suppose that $S$ is a non-compact hyperbolic surface of finite area. Suppose that $\lambda \subset S$ is a non-trivial, geodesic, measured lamination. ...
1
vote
1answer
348 views

Two questions from Hubbard's Teichmuller theory book Vol I, P. 130 , Thm 4.4.1, ( QC maps )

I was studying Theorem 4.4.1 from John H. Hubbard's Teichmuller Theory, vol I, Theorem 4.4.1 ( P. 129 ) which states : Let $X,Y$ be two hyperbolic Riemann surfaces with hyperbolic metrics $d_X,d_Y$ ...
3
votes
2answers
392 views

a little question about Heegaard splitting

for any compact orientable hyperbolic 3-manifold with totally geodesic boundary, is there a strongly irreducible heegaard splitting?
0
votes
1answer
273 views

Generator of translation for the hyperbolic plane? [closed]

What is the generator of translation in the Beltrami-Klein model of the hyperbolic plane?
2
votes
1answer
756 views

How people think about ending lamination?

There are lots of works in hyperbolic 3-manifolds related to ending lamination. But I just don't know how people think about it . what is the philosophy behind it? maybe i should ask how people ...
4
votes
1answer
579 views

A quick and elementary question from Hubbard's Teichmuller Theory : Volume I

Hi, On page 120, chapter 4, proposition 4.2.7 in Hubbard's Teichmuller Theory book, volume 1, he proves : Let $U,V$ be open in $C, f:U \to V $ be a homeomorphism and the restriction of $f$ on $U ...
12
votes
3answers
652 views

F→E→B bundle with B,E,F hyperbolic: possible?

It would be interesting to me obtain an answer to the following easy to state question: Does there exist a (smooth) fibre bundle $\pi\colon E\rightarrow B$ with typical fibre $F$ such that $E$, $B$ ...
4
votes
2answers
580 views

A construction of generators of discrete subgroups of SL(2,R)

I know about geometrical method of construction of discrete subgroups of $SL(2,\mathbb{R})$ using Lobachevsky plane (e.g. B.A. Dubrovin, A.T. Fomenko, S.P. Novikov, Modern Geometry --- Methods and ...
3
votes
6answers
620 views

Quasiconformal harmonic extension of a quasi-symmetric map on $S^1$

Hello ,we know that for given $h:S^1\to S^1$, we can solve the Dirichlet problem on $\bar{D} $ with the boundary value $h$ and in fact this extension, which is the complex harmonic extension $H=E(h) $ ...
3
votes
0answers
276 views

Boundary defining functions for hyperbolic surfaces

Let $M$ be a geometrically finite hyperbolic surface with one cuspidal end and one funnel end so that it can be divided into $C \cup K \cup F$ where $C$ is the cusp, $F$ the funnel and $K$ the ...
0
votes
1answer
158 views

Definition of k -quasisymmetric maps on S^1

I know the definition of k -quasi-symmetric maps $f$ on $R$,it is there exists $k>0$ such that $\frac{1}{k}\leq\frac{f(x+t)-f(x)}{f(x)-f(x-t)} \leq k \forall x,t\in R.$ So I just want to ...
3
votes
2answers
412 views

Capacity of Balls in Hyperbolic Space

Given $M$ a Riemannian manifold and $\Omega\subset M$ the capacity of $\Omega$ is defined as $$ \mathrm{cap}(\Omega)=\inf \int_{M\setminus\Omega}{|\mathrm{grad} \varphi|^2 dV} $$ where $\varphi$ ...
4
votes
1answer
347 views

Is there a concept of Combined Teichmuller space for surfaces with both geodesic boundary and punctures/cusps

If we take a sequence of compact hyperbolic Riemann surface with k geodesic boundary components such that the lengths of the geodesic boundary components go to zero, then in the "limit", we should get ...
12
votes
1answer
705 views

Gromov's Hyperbolicity and Positive Cheeger Constant in Planar Graphs

It is known that for metric graphs the concepts of Gromov's hyperbolicity and strictly positive Cheeger constant are related. Let us first recall the definition of the Cheeger constant. Let $G$ be a ...
5
votes
2answers
277 views

Subgroup structure of $\mathrm{SO}(1,n)_0$

A classification (up to conjugacy) of all closed subgroups of the (identity component of the) Lorentz group $\mathrm{SO}(1,3)_0$ in terms of the subalgebras of its Lie algebra was given in R. Shaw. ...
3
votes
2answers
537 views

Fundamental groups of closed hyperbolic 3-manifolds are freely indecomposable

I believe the following statement is true, and I've even seen it referenced here. Could someone point me to a proof? The fundamental group of a closed hyperbolic 3-manifold is not a free product. ...
0
votes
2answers
668 views

A quick question about Farb-Margalit's book on MCG's proof on Teichmuller's existence theorem

Hello, I was studying Farb-Margalit's " A Primer on MCG " for Teichmuller's existence theorem. On P. 347, proposition 11.14, they proved $ \omega : QD_1(X) -> Teich ( S_g) $ is proper, which, ...
1
vote
1answer
183 views

Connectedness of the thick part of a hyperbolic manifold?

In a solution to a recent post : Fundamental group of a thick part of hyperbolic manifold, Igor Belegradek makes this claim that the thick part of a hyperbolic manifold is connected. To me it seems ...
3
votes
2answers
300 views

Fundamental group of a thick part of hyperbolic manifold

Let $M$ be a complete hyperbolic manifold of dimension $n$, let $\varepsilon=\varepsilon_n$ be the Margulis constant. Let $M_{[\varepsilon,\infty)}$ be the thick part of $M$ with respect to ...
1
vote
1answer
148 views

Good references for Hyperbolic and parabolic annuli

I want to understand the geometry of the Hyperbolic annuli (Hyperbolic plane quoteinted by the group generated by $z\mapsto rz$ for a fixed $r$) and parabolic annuli (Hyperbolic plane quoteinted by ...
2
votes
1answer
227 views

Can we prove $ Aut(S_g) , g \geq 2 $ is finite in the following way ?

I was trying to prove that $ Aut( S_g $), g$ \geq 2 $ [ orientation preserving isometries ] is finite in the following way : For fixed $M $ ( positive ) there are finitely many , say $ k $ number of ...
1
vote
1answer
449 views

Some basic questions about the proof of Teichmuller's uniqueness theorem

Hello , I was studying the proof of Teichmuller's uniqueness theorem from the note/book " A Primer on Mapping Class Groups " by Farb-Margalit and I got struck at a couple of points, mainly because I ...
2
votes
1answer
309 views

invariant 2-form in hyperbolic 3-space

Hello all As is probably well-known to most, in the upper halfplane we have a natural action of $SL_2(\mathbb{R})$ through linear fractional transformations, and a $2$-form $\frac{dx dy}{y^2}$ which ...
13
votes
1answer
1k views

Detecting a cover of the figure-8 knot complement

I have a specific concrete example of a complete, finite volume, cusped hyperbolic 3-manifold $M$, and I am trying to determine whether or not $M$ is a cover of the figure-8 knot complement (call this ...
2
votes
2answers
292 views

Why a non-simple geodesic in a Y-piece is NOT homotopic to a common perpendicular to the geodesic boundaries ?

This is a basic question, still I dare to ask : Let Y be the Y-piece with geodesic boundaries A,B, C and ( if possible ) c the non simple geodesic from A to B intersecting itself at a point p. I want ...